Random Index from a Tensor (Sampling with Replacement from a Tensor) - python-3.x

Question

I'm trying to manipulate individual weights of different neural nets to see how their performance degrades. As part of these experiments, I'm required to sample randomly from their weight tensors, which I've come to understand as sampling with replacement (in the statistical sense). However, since it's high-dimensional, I've been stumped by how to do this in a fair manner. Here are the approaches and research I've put into considering this problem:
This was previously implemented by selecting a random layer and then selecting a random weight in that layer (ignore the implementation of picking a random weight). Since layers are different sizes, we discovered that weights were being sampled unevenly.
I considered what would happen if we sampled according to the numpy.shape of the tensor; however, I realize now that this encounters the same problem as above.
Consider what happens to a rank 2 tensor like this:
[[*, *, *],
[*, *, *, *]]
Selecting a row randomly and then a value from that row results in an unfair selection. This method could work if you're able to assert that this scenario never occurs, but it's far from a general solution.
Note that this possible duplicate actually implements it in this fashion.
I found people suggesting flattening the tensor and use numpy.random.choice to select randomly from a 1D array. That's a simple solution, except I have no idea how to invert the flattened tensor back into its original shape. Further, flattening millions of weights would be a somewhat slow implementation.
I found someone discussing tf.random.multinomial here, but I don't understand enough of it to know whether it's applicable or not.
I ran into this paper about resevoir sampling, but again, it went over my head.
I found another paper which specifically discusses tensors and sampling techniques, but it went even further over my head.
A teammate found this other paper which talks about random sampling from a tensor, but it's only for rank 3 tensors.
Any help understanding how to do this? I'm working in Python with Keras, but I'll take an algorithm in any form that it exists. Thank you in advance.

Answer 1

Before I forget to document the solution we arrived at, I'll talk about the two different paths I see for implementing this:
Use a total ordering on scalar elements of the tensor. This is effectively enumerating your elements, i.e. flattening them. However, you can do this while maintaining the original shape. Consider this pseudocode (in Python-like syntax):
def sample_tensor(tensor, chosen_index: int) -> Tuple[int]:
"""Maps a chosen random number to its index in the given tensor.
Args:
tensor: A ragged-array n-tensor.
chosen_index: An integer in [0, num_scalar_elements_in_tensor).
Returns:
The index that accesses this element in the tensor.
NOTE: Entirely untested, expect it to be fundamentally flawed.
"""
remaining = chosen_index
for (i, sub_list) in enumerate(tensor):
if type(sub_list) is an iterable:
if |sub_list| > remaining:
remaining -= |sub_list|
else:
return i joined with sample_tensor(sub_list, remaining)
else:
if len(sub_list) <= remaining:
return tuple(remaining)
First of all, I'm aware this isn't a sound algorithm. The idea is to count down until you reach your element, with bookkeeping for indices.
We need to make crucial assumptions here. 1) All lists will eventually contain only scalars. 2) By direct consequence, if a list contains lists, assume that it also doesn't contain scalars at the same level. (Stop and convince yourself for (2).)
We also need to make a critical note here too: We are unable to measure the number of scalars in any given list, unless the list is homogeneously consisting of scalars. In order to avoid measuring this magnitude at every point, my algorithm above should be refactored to descend first, and subtract later.
This algorithm has some consequences:
It's the fastest in its entire style of approaching the problem. If you want to write a function f: [0, total_elems) -> Tuple[int], you must know the number of preceding scalar elements along the total ordering of the tensor. This is effectively bound at Theta(l) where l is the number of lists in the tensor (since we can call len on a list of scalars).
It's slow. It's too slow compared to sampling nicer tensors that have a defined shape to them.
It begs the question: can we do better? See the next solution.
Use a probability distribution in conjunction with numpy.random.choice. The idea here is that if we know ahead of time what the distribution of scalars is already like, we can sample fairly at each level of descending the tensor. The hard problem here is building this distribution.
I won't write pseudocode for this, but lay out some objectives:
This can be called only once to build the data structure.
The algorithm needs to combine iterative and recursive techniques to a) build distributions for sibling lists and b) build distributions for descendants, respectively.
The algorithm will need to map indices to a probability distribution respective to sibling lists (note the assumptions discussed above). This does require knowing the number of elements in an arbitrary sub-tensor.
At lower levels where lists contain only scalars, we can simplify by just storing the number of elements in said list (as opposed to storing probabilities of selecting scalars randomly from a 1D array).
You will likely need 2-3 functions: one that utilizes the probability distribution to return an index, a function that builds the distribution object, and possibly a function that just counts elements to help build the distribution.
This is also faster at O(n) where n is the rank of the tensor. I'm convinced this is the fastest possible algorithm, but I lack the time to try to prove it.
You might choose to store the distribution as an ordered dictionary that maps a probability to either another dictionary or the number of elements in a 1D array. I think this might be the most sensible structure.
Note that (2) is truly the same as (1), but we pre-compute knowledge about the densities of the tensor.
I hope this helps.